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February 14, 2006, 09:26 
Re: Concept & idea consolidation

#101 
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diaw: I'm trying to understand why, in the pipe case, the solutions are isolated. This can certainly be said for traditional development of say gravity waves, where the perturbation is introduced, 3 sets of equations are developed  one for bulk flow, one for deviatoric & the final  for the nonlinear linkages. The approach then makes simplifying assumptions which essentially reduce the nonlinear linkage equation to zero  leaving two independent solutions operating in the same space  or drops the bulk solution completely  as in the case of gravity waves.
This is a standard property of nonlinear equations  solutions can exist which are not bifurcations from a common "base state". You really need to read about what was done to obtain these solutions and not look at a "popular" account which does not tell you all the facts. You probably also need some some understanding of nonlinear stability theory/bifurcation analysis to appreciate what the results actually mean. There are some nice colour pictures of these solutions in Wedin, H. & Kerswell, R.R. ``Exact coherent structures in pipe flow: travelling wave solutions'' J. Fluid Mech. 508, 333371, 2004. diaw: Granted, I am certainly referring to transient waveforms in the context of a transient wavefield. given a consistent boundary disturbance, would there be no possibility of nonlinear 'normal modes' forming? I think that perhaps the difference in viewpoints may be one of considering a reasonably consistent source of input 'noise' as a boundarycondition, or of flowgenerated noise/perturbations. I would love to know if sound measurements of the onset of instability have been conducted  I'm sure they have. Does flow in a pipe make a noise? In some cases, it certainly can. Most experimental fluid experiments use sound waves to excite the flow and induce transition. Turbulence does produce noise  just go outside on a windy day. There is a considerable amount of work on this  numerous papers by Lighthill, Crighton and Howe (Howe's also written a book vortex sound). To study this type of problems you must abandon the incompressible assumption and work with a fuller set of equations since the solenoidal condition filters all acoustic waves from the equations. There is no such thing as a nonlinear normal mode (by definition!) and even if there was there is no need to invoke it to describe the problem  the linear "branch cut" argument in the complex plane gives the solution for small amplitude disturbances (just as in the gravity wave case). diaw: What happens when we 'pull' flow out of a pipe  by setting a slight suction on the outlet boundary? (This one can be rather fun to observe, although some solvers will definitely not want to oblige.) In an incompressible fluid cavitation occurs (you are taking out more fluid than you are putting in) and the solution to the problem (as posed with noslip bcs) has failed to exist  hence the solvers that fail are the ones that are correct! For the solution to make sense you need to augment the statement of the problem with suitably altered boundary conditions. diaw:What would a pressurewave look like in a pipe? As I've already stated there is no pressure wave in the incompressible limit. 

February 14, 2006, 11:02 
Re: Concept & idea consolidation

#102 
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Thanks Tom for those comments. A fair bit to digest.
diaw: What happens when we 'pull' flow out of a pipe  by setting a slight suction on the outlet boundary? (This one can be rather fun to observe, although some solvers will definitely not want to oblige.) Tom: In an incompressible fluid cavitation occurs (you are taking out more fluid than you are putting in) and the solution to the problem (as posed with noslip bcs) has failed to exist  hence the solvers that fail are the ones that are correct! For the solution to make sense you need to augment the statement of the problem with suitably altered boundary conditions. diaw: I agree with cavitation occuring. I have a different perspective, however, on the role of the 'squishy' element & du/dx =  dv/dy & its role in allowing cavitationtype events & mass (momentum) redistribution to occur. I guess this will go back to the p=const vs div(V)=0 debate. diaw:What would a pressurewave look like in a pipe? Tom: As I've already stated there is no pressure wave in the incompressible limit. diaw: Perhaps again we disagree in terms of the role of du/dx =  dv/dy?  Concepts of singularities in flow fields: Could I ask what you views are on the possibilty of singularities existing in fluid flow fields? If so, what physical form would these take?  Physical approach (a full circle): When all is said & done, what I always fall back on, in terms of trying to understand the physical nature of flows is that, in general we have flows that meet the density=const requirement (as per my reference to Gas Dynamics), & that to all intents & purposes this can be at reasonably modest velocities. We do most definitely observe both wave activity & bulk activity coexisting together in such flows. When we started earlier discussions, we discussed gravity waves evident at the surface & I tried to draw down the pressure effects to an imaginary plane below the surface. A simple force balance would show that the gravity waves must be balanced by upthrusts from below  otherwise need to bridge from some distant river boundary, or obstacle  surface tension would merely convey the support from the distant boundary. Thus, to maintain equilibrium in the flow field at both surface & below, pressure fields must be active. In other words, see a pressure on the surface, you will be sure to see a pressure distribution below it on the cutplane, in some form. A cfd simulation will show pressure distributions on this cutplane that reasonably closely match the gravitywave phenomena on the surface. This is basic physics, as I see it. It must hold. Flows in musical instruments rely on pressure waves to do their job. Change the boundary condition, & we get a different wave structure & different sound. We have different harmonics etc. What happens when we move the fingers? Pressure waves operate. Does air move along the musical instrument (low bulk speed + coexistent wave)? There are so many examples around us of fluid phenomena that operate at very low velocities that exhibit wave & bulk phenomena. As far as engineers & physicists seem to be concerned, at these velocities (M<0.3 ~ 100 m/s!, for gases), the flows can be considered as density=constant. This makes no assumptions about the activity of du/dx + dv/dy = 0. It seems as though the mathematical & physics interpretations may be fundamentally at odds  even, though, in the end, we arrive at the same form of the continuity equation. The fact that infinite communication velocities cannot occur in practice, shows that something is incorrect in the physical outworking of the div(V)=0 approach. This worries me. Perhaps we have created restrictions on our perceptions that are preventing us from seeing a larger picture spreading from lowspeed to high speeds, where both waves (transient, persistent, nonlinear) & bulk flows are active. Could it be that we are perhaps not modeling nature in a completely physically meaningful way? In the end, if our equations do not allow us to arrive at answers that match the physics we observe around us, then we have really done something wrong. In my understanding of density=consant, with a 'squishy' (deformable) element, I'm still not convinced that pressure waves cannot move back & forth until standingwave structures set themselves in place. I know that we have discussed the implications of the mathematical interpretation of these flows, but to my observeration, they don't ring true. In my view, we really need a physical paradigm shift to take us away from this viscous, lowspeed, 'incompressible' trap. If not wave+bulk field interaction, then perhaps we can develop an alternative understanding  but, it should not resort to the idea of turbulence being a statistical concept. This cannot be correct  nature is not that way  it may perhaps have much more to do with how we interpret nature. Nature is simple  yet complex. When we begin to become overly complex in our understanding then I vote that it may be time to rethink the physics. When we can explain & model the Reynold's experiment in a pipe  then we may be in the right ballpark.  The elegance of the dualnature of the NS  the 'hidden nature' & a coexistent nonlinear models, with presure interlinking may go some way to answering some of these questions. I will read every link you have so kindly provided  this is a blessing indeed. I also plan to devote my time to gaining a complete understanding of the bridging paradigm we so desperately need. I plan to build the numerical models in a few different environments to test out the concepts. Lets see where it all goes.  Tom, thanks so much for your extremely kind debate & for making me a lot wiser than when we first began. You have been wonderfully patient, kind & generous  thank you. You are indeed a master craftsman. Des Aubery... 

February 15, 2006, 23:39 
Wavesolutions & div(v) revisited

#103 
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Wavesolutions & div(v) revisited
During the last few days, after the previous discussions, I settled a number of research firmly issues in my mind. I would draw your attention to an excellent book, "The Physics of Vibrations & Waves", Pain H.J., 6th Ed, Wiley, 2005 (1) Incompressibility & dilation For a constant mass, Compressibility: B =  dP/(dV/V) Incompressible => B > infinity; => dilation = (dV/V) > 0  (2) Deformable fluid element Relationship between longitudinal compression & lateral distortion Proceed with derivation as per 'solid' constant mass element. With a few observations, the div(v) = du/dx + dv/dy = 0 emerges. Proceeding along the development of elastic waves in solids is extremely informative. Conclusion: The deformable 'incompressible' fluid element meets both dilation = 0, div(V)=0 requirements. This element supports wave activity.  (3) Momentum (velocity) & pressure waves in NS The full nonlinear form of the incompressible NS equations admit a solution of the form: () = ()o.e^(at).e^i(kx + ly b.wt) where: a ~ visc*K^2 ; b ~ fn(()o.e^(at).e^i()) (nonlinear) At this point, no simplifying assumptions are made  no linearisation etc. The main topic of interest to note is the term e^(kvisc*K^2.t), which, in most practical terms (kvisc~855e9) would tend to e^(~0)~1. This admits the wave solution in any time window within reason. It is impractical to consider the limit t> infinity, as this would mean that all experiments would take longer than the average lifetime. For all practical intents & purposes, as suitable time for observation should be selected as the 'steadylimit'. The 'b' finding implies that 'b' itself admits wavesolutions, & thus the timing within the solution constantly moves around as time advances. The e^(0)~1 decay term, to all intents & purposes, admits a 'longterm' standingwave solution. I have observed these effects at scales, in many models, which are definitely not in the region of numericroundofferror.  With this latter research clarification, I am now comfortable with the 'transient wavesolution' to the NS & understand the role it plays. It appears that the problem may not purely lie in numeric roundoff, but in the NS solutions themselves.  Can I respectfully ask advice on suitable Journals to consider when publishing my research findings? Thanks again, diaw... (Des Aubery) 

February 16, 2006, 05:32 
Re: Wavesolutions & div(v) revisited

#104 
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"Can I respectfully ask advice on suitable Journals to consider when publishing my research findings?"
If you really believe you have something new and interesting to say about the equations of motion then you should consider either the Journal of fluid mechanics, Proceedings of the Royal society of London A, European Journal of Mechanics B/fluids, physics of fluids or possibly Theroetical and computational fluid dynamics. However I would suggest that, if you submit to one of these journals, you prepare yourself for some harsh critism/rejection (JFM rejects around 55% of submitted papers as policy!). It is always worth the extra effort to publish in a respected journal though. 

February 16, 2006, 06:44 
Re: Wavesolutions & div(v) revisited

#105 
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Tom:
"Can I respectfully ask advice on suitable Journals to consider when publishing my research findings?" If you really believe you have something new and interesting to say about the equations of motion then you should consider either the Journal of fluid mechanics, Proceedings of the Royal society of London A, European Journal of Mechanics B/fluids, physics of fluids or possibly Theroetical and computational fluid dynamics. However I would suggest that, if you submit to one of these journals, you prepare yourself for some harsh critism/rejection (JFM rejects around 55% of submitted papers as policy!). It is always worth the extra effort to publish in a respected journal though. diaw: Thanks so much Tom, for the Journal list. Those are indeed very well respected Institutions. I would certainly expect some very harsh criticism/rejection  this would most certainly be in order. I believe that this would be well worth the effort. In some ways, your gentle wise comments during the course of this debate have been preparing my mental resolve. At each point, it has sent me more & more deeply into the books to try & explain what I have seen. This has forced me to go back to basics in some areas & explain my case more thoroughly. No doubt, the review team of some of these Journals will put my research through the distilling fires  but, so, let it be. Once again, thank you so much for your debate & wisdom. It has been suncerely appreciated. Des Aubery... 

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